Probability Question

Question

A multiple choice exam consists of 25 questions, each with 4 possible answers. Suppose that a student doesn't know the material and simply guesses on each question.

What is the probability that they get more than 3 but less than 10 correct?

Answer 1

Hello Mika! How are you?

We want the probability of getting 4 to 9 questions right.

The result of each questions is independent of the other. The probability of getting one right by chance is:

p = 1/4 = 0.25

The probability of getting two right, by chance is:

p*p = p² = 0.0625

The probability of getting more than 3 and less than 10 is:

P = p⁴ + ...+ p⁸ + p⁹

P = p⁴*(1 + p + ... + p⁵ + p⁵) = p⁴*(p⁶ -1)/(p-1) (from the sum of a geometrical progression)

Note that:

(p⁶ -1) = (p³ + 1)*(p³ - 1)

(p³ - 1) = (p - 1)*(p² - p + 1)

then

(p⁶ -1) = (p³ + 1)*(p² - p + 1)*(p - 1)

We can then rewrite:

P = p⁴*(p³ + 1)*(p² - p + 1) I cancel (p - 1)

P = 4^-4*(1/64 +1)*(1/16 -1/4 +1)

P = 4^-4*(65/64)*(13/16)

(sorry for the ugly expression, I really tried not to use a calculator)

Feel free to send me a message, ok? Best regards!